American Institute of Mathematical Sciences

Advanced
June  2014, 6(2): 261-278. doi: 10.3934/jgm.2014.6.261

Periodic orbits in the Kepler-Heisenberg problem

Corey Shanbrom 1,

1. 

Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA, United States

Received  November 2013 Revised  March 2014 Published  June 2014

One can formulate the classical Kepler problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. Here we use variational methods to prove that the Kepler-Heisenberg system admits periodic orbits with $k$-fold rotational symmetry for any odd integer $k\geq 3$. Approximations are shown for $k=3$.
Keywords: Heisenberg group, Periodic orbits, fundamental solution to Laplacian., integrable system, Kepler problem, Carnot group.
Mathematics Subject Classification: 53C17, 37N05, 37J45, 70H12, 70H0.
Citation: Corey Shanbrom. Periodic orbits in the Kepler-Heisenberg problem. Journal of Geometric Mechanics, 2014, 6 (2) : 261-278. doi: 10.3934/jgm.2014.6.261
References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978. Google Scholar

[2]

A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2. doi: 10.1134/S156035470806004X.  Google Scholar

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.  Google Scholar

[4]

D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416.  Google Scholar

[5]

G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744. doi: 10.2307/2370714.  Google Scholar

[6]

O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960.  Google Scholar

[7]

R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27.  Google Scholar

[8]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.  Google Scholar

[9]

F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266. doi: 10.1007/s00332-011-9116-z.  Google Scholar

[10]

G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4.  Google Scholar

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963.  Google Scholar

[12]

W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.  Google Scholar

[13]

P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983.  Google Scholar

[14]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow. Google Scholar

[15]

R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002. Google Scholar

[16]

R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere,, to appear in Geometry, ().   Google Scholar

[17]

R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.  Google Scholar

[19]

C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013.  Google Scholar

[20]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980. Google Scholar

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978. Google Scholar

[2]

A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2. doi: 10.1134/S156035470806004X.  Google Scholar

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978.  Google Scholar

[4]

D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416.  Google Scholar

[5]

G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744. doi: 10.2307/2370714.  Google Scholar

[6]

O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960.  Google Scholar

[7]

R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27.  Google Scholar

[8]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901. doi: 10.2307/2661357.  Google Scholar

[9]

F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266. doi: 10.1007/s00332-011-9116-z.  Google Scholar

[10]

G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376. doi: 10.1090/S0002-9904-1973-13171-4.  Google Scholar

[11]

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963.  Google Scholar

[12]

W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.  Google Scholar

[13]

P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983.  Google Scholar

[14]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow. Google Scholar

[15]

R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002. Google Scholar

[16]

R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere,, to appear in Geometry, ().   Google Scholar

[17]

R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30. doi: 10.1007/BF01941322.  Google Scholar

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.  Google Scholar

[19]

C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013.  Google Scholar

[20]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980. Google Scholar

[1]

Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775
[2]

Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113
[3]

Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209
[4]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[5]

L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107.

[6]

Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
[7]

Luis F. López, Yannick Sire. Rigidity results for nonlocal phase transitions in the Heisenberg group $\mathbb{H}$. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2639-2656. doi: 10.3934/dcds.2014.34.2639
[8]

Patrizia Pucci. Critical Schrödinger-Hardy systems in the Heisenberg group. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 375-400. doi: 10.3934/dcdss.2019025
[9]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779
[10]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109
[11]

Carlota Rebelo, Alexandre Simões. Periodic linear motions with multiple collisions in a forced Kepler type problem. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3955-3975. doi: 10.3934/dcds.2018172
[12]

Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053
[13]

Houda Mokrani. Semi-linear sub-elliptic equations on the Heisenberg group with a singular potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1619-1636. doi: 10.3934/cpaa.2009.8.1619
[14]

Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29 (4) : 2673-2685. doi: 10.3934/era.2021008
[15]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[16]

Mohamed Baouch, Juan Antonio López-Ramos, Blas Torrecillas, Reto Schnyder. An active attack on a distributed Group Key Exchange system. Advances in Mathematics of Communications, 2017, 11 (4) : 715-717. doi: 10.3934/amc.2017052
[17]

Mamadou L. Diagne, Ousmane Seydi, Aissata A. B. Sy. A two-group age of infection epidemic model with periodic behavioral changes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2057-2092. doi: 10.3934/dcdsb.2019202
[18]

Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
[19]

Chengxin Du, Changchun Liu. Time periodic solution to a two-species chemotaxis-Stokes system with $ p $-Laplacian diffusion. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021162
[20]

Tzu-Hsin Liu, Jau-Chuan Ke, Ching-Chang Kuo. Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1411-1422. doi: 10.3934/jimo.2020027

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences